On the P\'olya conjecture for circular sectors and for balls

TL;DR

This paper proves Polya's conjecture for eigenvalue counting functions in circular sectors and for balls in any dimension, confirming the conjecture's validity in these geometries.

Contribution

It establishes the conjecture for circular sectors and for balls of arbitrary dimension, extending previous partial results.

Findings

Proved Polya's conjecture for circular sectors.

Confirmed the conjecture for balls in any dimension.

Applied methods from previous research to new geometries.

Abstract

In 1954, G. Polya conjectured that the counting function $N(\Omega,\Lambda)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset R^d$ is lesser (resp. greater) than $(2\pi)^{-d} \omega_d |\Omega| \Lambda^{d/2}$. Here $\Lambda$ is the spectral parameter, and $\omega_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.